Benign Projective Landscapes for Measured Quantum Divergences

Abstract

We study nonconvex optimization of measured quantum $f$-divergences over rank-one projective measurements. For smoothly operator-Fenchel liftable generators and faithful states, every projective local maximum and every second-order stationary point is globally optimal over all POVMs. The criterion is blockwise: a critical PVM is optimal exactly when the compressed states are proportional on each equal-score block; otherwise an explicit two-vector rotation has positive ascent curvature. Operator-convex generators admit a positive atomic curvature resolution, and the quadratic $\chi^2$ case yields two-sided residual bounds. For binary accessible information, this framework proves the known adaptive-capacity equality and shows that every nonidentical qubit ensemble has exactly two stationary projective measurements, proving conjectures of Keil and Thai--Dall'Arno. A rare-prior limit connects weighted Jensen--Shannon information to relative entropy and yields a finite counterexample to the proposed equivalence between observational-entropy and all-ensemble mutual-information orders. The landscape theorem also covers measured R\'enyi divergences of finite positive order and measured relative entropy.